Page 13 - Textos de Matemática Vol. 36

1.1. Riccati-transformation 3
This means that the stationary point −i ”attracts” all initial values represented with α ̸= 0. In other words we conclude from this observation that the comple- ment of the ”basin of attraction (in Cˆ)” of −i has measure 0. In particular the
basin of attraction of −i is open and dense in Cˆ). An analogous conclusion can be drawn for the second stationary point +i. This also reveals that all solutions of the algebraic Riccati equation can be obtained as a limit from a solution of the matrix Riccati differential equation.
This example already shows many of the features of the solutions of general matrix Riccati equations.
Before we discuss the representation of these solutions we present in this introductory part some applications of matrix Riccati differential equations and also of algebraic Riccati equations
First we present the Riccati-transformation to obtain a spectral factoriza- tion of a linear matrix operator.
1.1 Riccati-transformation
In singular perturbations (see [Waso95], [OMal74]) and also in many other fields, as for example in control theory (see [KOS76], [AnMo90], [AgGa93], [SuGa91], [Frid95]), one frequently uses special linear transformations in or- der to reduce high-order systems to lower order ones or in order to (partially) decouple the system.
If the original linear system x˙ = A(t)x + B(t)u is partitioned as x˙ 1 A11 (t) A12 (t) x1 B1 (t)
(1.1)
(1.2) denotes a
x˙
= A (t) A (t) x + B (t) u, 2 21 22 2 2
where A11(t) ∈ Rn×n, A12(t) ∈ Rn×m, A21(t) ∈ Rm×n, A22(t) ∈
B1(t) ∈ Rn×k and B2(t) ∈ Rm×k, it may be conveniently transformed into decoupled subsystems by a two step transformation.
In the first step we use the transformation
In T(t) := K(t)
−1 In (t) = −K(t)
with the inverse T
solution of the non-symmetric Riccati differential equation RDE
0 (m+n)×(m+n)
∈ R
0 m×n
I
I and where K(t) ∈ R
m
K˙ =−KA11(t)+A22(t)K−A21(t)+KA12(t)K.
m
Rm×m,
(1.3)